This topic came up on a different forum recently, with some really persuasive writing and math supporting the possibility of the tail providing lift [nose down direction] in some C of G configurations. I'm not persuaded yet.
To support the theme that Cobalt and others have presented, if there were a configuration where in stable flight the tail lifted up, rather than down, as convention has it, how would you rotate for takeoff in a tricycle aircraft? The tail would already be on the ground! A Cessna Caravan, as other types has a "pogo stick" for loading when parked, but not 'cause you need it for taxi takeoff and landing!
I remain convinced that in sable flight, in a conventional aircraft, the tail always provides a downforce to some degree. I have flown Cessnas a bit behind the aft C of G limit, and they still required a nose up control input to rotate on takeoff, because they did not rotate themselves - happily!